Conformal transformations of Riemannian manifolds

Obata, Morio

doi:10.4310/jdg/1214429505

Cited by 87 publications

(74 citation statements)

References 15 publications

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“…The main result is the same as in the conformai case [6]. This is used to show that, if M is as in the preceding paragraph and has a finite fundamental group, then M is globally equivalent to S2n+X.…”

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confidence: 67%

On the transformation group of a real hypersurface

Webster¹

1977

Trans. Amer. Math. Soc.

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Abstract. The group of biholomorphic transformations leaving fixed a strongly pseudoconvex real hypersurface in a complex manifold is a Lie group. In this paper it is shown that the Chern-Moser invariants must vanish if this group is noncompact and the hypersurface is compact. Also considered are transformation groups of flat hypersurfaces and intransitive groups.Introduction. The purpose of this paper is to study the group of structure preserving, or pseudoconformal, transformations of a real hypersurface M (of dimension 2n + 1) in a complex n + 1 manifold, or more generally, of a manifold with the same C-R structure. We will always assume that the C-R structure is integrable and that the Levi form is nondegenerate. It follows from the work of Chern and Moser [2] that this group is a Lie transformation group.There is some similarity between the geometry of M and the conformai geometry of a Riemannian manifold. Over M there is a principal fibre bundle B (the pseudoconformal bundle), a connection on B, and curvature invariants the most important of which is the fourth order curvature tensor Sg^ (see [2]). These have as analogues in the Riemannian case the bundle of conformai frames the conformai connection, and the Weyl conformai curvature tensor. If n > 2, S vanishes if and only if M is pseudoconformally flat, i.e., locally equivalent to the standard sphere S2n+l in C+I [8].Locally the structure of M is given by a real one-form 9, n complex one-forms 9a, and their complex conjugates 9" -9a. They satisfy

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confidence: 67%

On the transformation group of a real hypersurface

Webster¹

1977

Trans. Amer. Math. Soc.

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“…We also make use of the substantial simplifications of Li and Zhang in [18] to the proof in [19]. The proof is along the line of the pioneering work of Gidas, Ni and Nirenberg [8], which in particular does not need the kind of divergence structure needed for the method of Obata [20] and therefore can be applied in much more generality.…”

Section: Theorem 4 ([15])mentioning

confidence: 96%

“…More recently, Guan, Viaclovsky and Wang [9] have proved that λ(A g ) ∈ Γ k for k ≥ n 2 implies the positivity of the Ricci tensor, and therefore, by classical results, (M, g) is conformally covered by S n and the existence and compactness results in this case follow easily. Our proof of Theorem 1, different from the ones in [20], [8], [3], [28] and [29], is in the spirit of the new proof of the Liouville type theorem of Caffarelli, Gidas and Spruck given by Li and Zhu in [19]. We also make use of the substantial simplifications of Li and Zhang in [18] to the proof in [19].…”

Section: Theorem 4 ([15])mentioning

confidence: 99%

On some conformally invariant fully nonlinear equations

2003

Comm Pure Appl Math

180

296

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We will report some results concerning the Yamabe problem and the Nirenberg problem. Related topics will also be discussed. Such studies have led to new results on some conformally invariant fully nonlinear equations arising from geometry. We will also present these results which include some Liouville type theorems, Harnack type inequalities, existence and compactness of solutions to some nonlinear version of the Yamabe problem.2000 Mathematics Subject Classification: 35, 58.

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“…If Sg denotes the scalar curvature of the metric g, one then has the classical formulas (see [14] for details)…”

Section: 9mentioning

confidence: 99%

“…IV.1 We pointed out in lI.ll that the map F2 intertwines the actions 02 The counterimage of Sc = n(n -1) by Fn is reduced to the On-orbit of the constant function 1 if n 2: 3 or 0 if n = 2 as follows from a theorem of M. Obata (cf. [14]), namely…”

Section: Iii2 By Its Very Definition the Set Of Forbidden Functionsmentioning

confidence: 99%

Scalar curvature functions in a conformal class of metrics and conformal transformations

Bourguignon¹,

Ezin²

1987

Trans. Amer. Math. Soc.

102

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ABSTRACT. This article addresses the problem of prescribing the scalar curvature in a conformal class. (For the standard conformal class on the 2-sphere, this is usually referred to as the Nirenberg problem.) Thanks to the action of the conformal group, integrability conditions due to J. L. Kazdan and F. W. Warner are extended, and shown to be universal. A counterexample to a conjecture by J. L. Kazdan on the role of first spherical harmonics in these integrability conditions on the standard sphere is given. Using the action of the conformal groups, some existence results are also given.O. Statements of results. Recently, the study of functions which, on a connected Coo n-manifold M without boundary, are scalar curvatures of complete Riemannian metrics has drawn special attention among both analysts and geome- Lawson (cf. [6]). Nevertheless, one among the oldest questions related to scalar curvature functions remains unsolved, the Nirenberg problem, namely "describe all curvature functions on the 2-sphere conformal to a standard one." In this article, we shall address this problem. Since it concerns the most familiar compact manifold 8 2 with its standard conformal class, it was expected to be a simple case of the more involved problem: "On a complete n-dimensional manifold, describe the scalar curvature functions in a given conformal class." (For recent developments on the subject, see [2, 4 and 5] [8]), J. L. Kazdan and F. Warner showed that, even if they were positive somewhere (as made necessary by the Gauss-Bonnet theorem), monotonic functions of the distance to a point were forbidden as curvature functions. This followed from an identity satisfied by first spherical harmonics on 8 2 , which, when correctly stated, generalized to the n-sphere 8 n . More recently, T. Aubin pushed further sharp Sobolev estimates for constrained functions by considering certain submanifolds of the Sobolev space H 1 (M). In this way, in [1], he

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Conformal transformations of Riemannian manifolds

Cited by 87 publications

References 15 publications

On the transformation group of a real hypersurface

On the transformation group of a real hypersurface

On some conformally invariant fully nonlinear equations

Scalar curvature functions in a conformal class of metrics and conformal transformations

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